In the Monty Hall problem there are three doors. Two of them conceal goats and the third has a fabulous prize behind it. The goal is to choose the door with the prize.

You make your choice and then Monty opens one of the remaining two doors. The rule is that Monty must always open a door with a goat behind it.

At this point there are two unopened doors, the one you chose and the one Monty didn't open. You are now given the opportunity to switch to the other door. Should you do it?

An incredible number of words, equations, diagrams, etc. have been spilled on this question. For an exhaustive description of the problem along with references see the Wikipedia article.

There is even a book on the subject.

The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brainteaser, by Jason Rosenhouse

In short, the answer to the question is yes, you should always switch. What follows is, I hope, the shortest and clearest explanation of why.

If you use the strategy of always switching then you are guaranteed to win the prize if your initial choice is a goat. This is because Monty always opens a door with a goat and if your choice is also a goat then the remaining door must be the prize.

The probability of you initially choosing a goat is 2/3 therefore this is the probability of winning by switching. You win by not switching only if your initial choice was the prize but the probability of this is only 1/3.

You therefore double your probability of winning by switching.

© 2010-2012 Stefan Hollos and Richard Hollos

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